How is the performance of a model affected by adding a ReLU to fully connected layers?
ReLU is piecewise linear function that outputs the received input directly if it's positive, or outputs a zero. i.e., $max(0, x)$

How significant is adding relu to full connected layers?
ReLU, being an activation function, will determine what the output of the nodes in your FCs are. Since it's a non-linear function, one significance is it will allow the nodes in your model to learn  complex mappings between the inputs and the outputs. Compared to using a linear function, it will allow back-propagation since it has a derivative, allowing the neural network to have the advantage of stacking of multiple FC layers.
Is it necessary?
ReLU (and non-linear activation functions in general) will introduce non-linear properties in a neural network that enables it to learn more complex arbitrary structures in the inputs. Without activation functions between the layers, your neural network will simply be a linear function, regardless of the number of layers it has. Why? A linear function + linear function gives a linear function. Additionally, see this answer.
How is the performance affected by adding ReLU?
Compared to no activation function at all, it will be slower to train but will only behave like a linear regression model. So ReLU will increase the power of the model by making it non-linear.
However, compared to other non-linear activation functions like tanH, ReLU will speed up training as (1) its computation step is cheaper i.e., $0.0$ or $x$ without additional operations (2) its gradient just depends on the sign of the input $x$. See this answer